Use the given row transformation to change each matrix as indicated. [ 1 4 4 7 ] , − 4 × row 1 added to row 2 \left[ \begin{matrix} 1 & 4 \\ 4 & 7 \end{matrix} \right], \quad -4 \times \text{row 1 added to row 2}

Hello, everyone. We are asked to apply the indicated row transformation to the following matrix. The matrix we are given is a two by two matrix where the first row is 38 and the second row is 85. We are asked to add the result negative five times of row one to row two. We have four answer choices all of which are two by two matrices with slightly different signs on the values of the elements. Thinking about what add the result negative five times row one to row two means this means we're going to do negative five multiplied by the element from row one and then added to the element from row two. So row one will not change starting a new matrix that stays three and eight. Nothing's changing there in row two. I'm gonna do negative five and I'm gonna multiply it by the corresponding value from row one. So the first element from row one, which is three and add that to the first element from row two, which is eight. I'm gonna do the same thing with the second element in row two. I'm gonna do negative five multiplied by the second element from row one, which is eight plus the second element from row two, which is five. And now to do the multiplication. In addition, again, our first row is still three and eight that did not change our second row negative five multiplied by three is negative plus eight is negative seven. For the second element negative five multiplied by eight is negative 40 plus a positive five is negative 35. So our answer is a two by two matrix where the first row is 38 and the second row is negative seven, negative 35. And this matches answer choice D have a nice day.

7. Systems of Equations & Matrices

Introduction to Matrices

College Algebra

7. Systems of Equations & Matrices

Introduction to Matrices

2:27 minutes

Problem 7

Textbook Question

Use the given row transformation to change each matrix as indicated.
$[\begin{matrix} 1 & 4 \\ 4 & 7 \end{matrix}],$

Verified step by step guidance

Identify the given matrix as a 2x2 matrix with elements arranged as: $\begin{bmatrix} a & b \\ c & d \end{bmatrix}$, where the first row is $(a, b)$ and the second row is $(c, d)$.

Understand the row operation: '-4 times row 1 added to row 2' means you multiply each element of the first row by -4 and then add the result to the corresponding element in the second row.

Calculate the new second row elements using the formula: $\text{new row 2} = \text{row 2} + (-4) \times \text{row 1}$, which translates to $\begin{bmatrix} c + (-4) \times a & d + (-4) \times b \end{bmatrix}$.

Keep the first row unchanged, so the first row remains $\begin{bmatrix} a & b \end{bmatrix}$.

Write the resulting matrix after the row operation as $\begin{bmatrix} a & b \\ c - 4a & d - 4b \end{bmatrix}$.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Matrix Representation and Notation

A matrix is a rectangular array of numbers arranged in rows and columns. Understanding how to read and write matrices, especially 2x2 matrices, is essential for performing operations like row transformations. Each element is identified by its row and column position.

Elementary Row Operations

Elementary row operations include swapping rows, multiplying a row by a scalar, and adding a multiple of one row to another. These operations are used to simplify matrices or solve systems of equations. In this problem, adding -4 times row 1 to row 2 modifies the second row accordingly.

Performing Row Transformations

To perform a row transformation, multiply each element of the specified row by the given scalar and add the result to the corresponding element of the target row. This changes the target row while leaving others unchanged, helping to achieve matrix simplification or row echelon form.

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